Entanglement and boundary entropy in quantum spin chains with arbitrary direction of the boundary magnetic fields

Abstract

We calculate the entanglement and the universal boundary entropy (BE) in the critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory (BCFT) depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, that apply to the ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group (DMRG). The transverse field XY chain with ADBMF after Jordan-Wigner (JW) transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem {\it{integrable}}. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.